Small Samples

Distributions

Power

Independent Random Variables

When a study involves pairs of
random variables,
it is often useful to know
whether or not the random variables are independent. This lesson
explains how to assess the independence of random variables.

Joint Probability Distributions

In a joint probability distribution
table, numbers in the cells of the table represent the probability
that particular values of X and Y occur together. From this table,
you can see that the probability that X=0 and Y=3 is 0.1; the probability
that X=1 and Y=3 is 0.2; and so on.

You can use tables like this to figure out whether two discrete
random variables are independent or dependent. Problem 1 below
shows how.

Test Your Understanding

Problem 1

The table below shows the joint probability distribution between
two random variables - X and Y.

X

0

1

2

Y

3

0.1

0.2

0.2

4

0.1

0.2

0.2

And the next table shows the joint probability distribution between two random variables - A
and B.

A

0

1

2

B

3

0.1

0.2

0.2

4

0.2

0.2

0.1

Which of the following statements are true?

I. X and Y are independent random variables.
II. A and B are independent random variables.

(A) I only
(B) II only
(C) I and II
(D) Neither statement is true.
(E) It is not possible to answer this question, based on the
information given.

Solution

The correct answer is A. The solution requires several computations
to test the independence of random variables. Those computations are
shown below.

X and Y are independent if P(x|y) = P(x), for all values of X and Y.
From the probability distribution table, we know the following: